Step by step
The question
Question: dogs: 40 walk, 10 run; cats: 25 walk, 25 run: at the 5% level, is the answer linked to the group?
1 Write the hypotheses
H₀ is the "nothing new" idea.
Here: the group and the answer are independent.
Hₐ is what we want to show.
Here: the group and the answer are not independent.
Independent means no link: the answer does not depend on the group.
2 Totals
The counts in a table.
Each row's total: dogs, 50; cats, 50.
Each column's total: walk, 65; run, 35.
And everyone: 100.
3 Expected counts
If there is no link, each cell expects its row total times its column total, over the grand total.
Dogs, walk: 50 times 65, over 100, is 32.5.
Dogs, run: 50 times 35, over 100, is 17.5.
Cats, walk: 50 times 65, over 100, is 32.5.
Cats, run: 50 times 35, over 100, is 17.5.
4 Each part
Chi-square adds up, for each cell, the observed minus the expected, squared, over the expected.
A table: each cell, what we saw, O, and what we expected, E.
40 minus 32.5 is 7.5.
Squared, 56.25.
Over 32.5: 1.7308.
10 minus 17.5 is −7.5.
Squared, 56.25.
Over 17.5: 3.2143.
25 minus 32.5 is −7.5.
Squared, 56.25.
Over 32.5: 1.7308.
25 minus 17.5 is 7.5.
Squared, 56.25.
Over 17.5: 3.2143.
5 Add them
1.7308 plus 3.2143 plus 1.7308 plus 3.2143 is 9.8902.
6 The cut-off
df is rows minus 1, times columns minus 1: 1 times 1 is 1.
Careful!
Not the number of cells minus 1.
Rows minus 1, times columns minus 1: 1.
Chi-square is always a right-tail test: big means far from what we expected.
Row 1, column 0.05.
Row 1, column .05: the cut-off is 3.841.
7 Decide
The chi-square curve for df 1.
Past the cut-off, 3.841, is the reject part.
9.8902 is past the cut-off.
So we reject H zero.
8 Say it in words
In words: at the 5% level, there is enough evidence that the group and the answer are not independent.
9 Check
Let's check: the expected counts give the same row totals.
Dogs: 32.5 plus 17.5 is 50.
Cats: 32.5 plus 17.5 is 50.
So it is right!
Let's look back.
First: write the hypotheses.
Then: totals.
Then: expected counts.
Then: each part.
Then: add them.
Then: the cut-off.
Then: decide.
Then: say it in words.
Last: check.